Forced convection in a helical pipe filled with a saturated porous medium

An analysis is made of laminar forced convection in a helical pipe of circular cross-section and filled by a porous medium saturated with a fluid, for the case when the curvature and torsion of the pipe are both small. The Darcy model is employed, and boundaries with either uniform flux or uniform temperature are considered. It is found that curvature induces a secondary flow at first order in the parameter ε = κa, where κ is the curvature and a is the radius of the pipe. On the other hand, the Nusselt number is unchanged to first order in ε but is increased at second order, for either set of thermal boundary conditions. The effect of torsion on the velocity appears at second order, but torsion does not affect the Nusselt number at second order.     

A thermal resistance analysis on forced convection with viscous dissipation in a porous medium using entransy dissipation concept

The analytical solution of a two-equation model presented in an earlier study is examined. Heat transfer characterization is classified into two regimes which are dominated by fluid conduction or solid conduction and interstitial heat exchange, respectively by using the entransy dissipation concept. The computed pattern of variation of thermal resistance with shape factor S at a fixed Brinkman number for a low ratio of the fluid to solid effective thermal conductivities implies the occurrence of temperature gradient bifurcation as S decreases. Therefore, the thermal diffusion term in the fluid phase in the two-equation model is not negligible for both regimes.     

A bioheat transfer model: Forced convection in a channel occupied by a porous medium with counterflow

An illustrative model for bioheat transfer is developed. An analytical solution is obtained for forced convection in a parallel plate channel occupied by a layered saturated porous medium with counterflow, the dominant feature that distinguishes bioheat transfer from other forms of heat transfer. The case of asymmetrical constant heat-flux boundary conditions is considered and the Brinkman model is employed for the porous medium. It is found that the Nusselt number Nu is zero when the mean velocity is zero, and negative values can be attained.     

Forced convection in porous microchannels with viscous dissipation in local thermal non-equilibrium conditions

Fully developed, steady-state forced convection, in parallel-plate microchannels, filled with a porous medium saturated with rarefied gases at high temperatures, in local thermal non-equilibrium (LTNE) condition, is investigated for the first-order slip-flow regime (0 ≤ Kn ≤ 0.1). Both velocity and temperature jumps at the walls are accounted for. An analytic solution is proposed for the Darcy–extended Brinkman flow model with assigned uniform heat flux at the microchannel walls and viscous heat dissipation in the fluid phase. The solution for NTLE includes the shear work done by the slipping effects. A closed-form expression of the Nusselt number is derived. A validation analysis with respect to the case of channels filled with saturated porous medium is accomplished. The results show that the internal dissipation increases as the velocity slip increases. In addition, the heat dissipation strongly affects the fluid temperature profiles. The increases in velocity slip and temperature jump lead to decreases of temperature gradients in the fluid and solid along the sections. The heat transfer at channel walls is enhanced due to an increase in the bulk heat transfer.     

On natural convection in enclosures filled with fluid-saturated porous media including viscous dissipation

Care needs to be taken when considering the viscous dissipation in the energy conservation formulation of the natural convection problem in fluid-saturated porous media. The unique energy formulation compatible with the First Law of Thermodynamics informs us that if the viscous dissipation term is taken into account, also the work of pressure forces term needs to be taken into account. In integral terms, the work of pressure forces must equal the energy dissipated by viscous effects, and the net energy generation in the overall domain must be zero. If only the (positive) viscous dissipation term is considered in the energy conservation equation, the domain behaves as a heat multiplier, with an heat output greater than the heat input. Only the energy formulation consistent with the First Law of Thermodynamics leads to the correct flow and temperature fields, as well as of the heat transfer parameters characterizing the involved porous device. Attention is given to the natural convection problem in a square enclosure filled with a fluid-saturated porous medium, using the Darcy Law to describe the fluid flow, but the main ideas and conclusions apply equally for any general natural or mixed convection heat transfer problem. It is also analyzed the validity of the Oberbeck–Boussinesq approximation when applied to natural convection problems in fluid-saturated porous media.     

A three-velocity three-temperature model for a tridisperse porous medium: Forced convection in a channel

A three-velocity three-temperature model for tridisperse porous media is formulated. Using the model, an analytic solution is obtained for the problem of forced convection in a channel between parallel plane walls that are held either at uniform temperature or uniform heat flux. In each case, Nusselt number values are given as functions of conductivity ratios, velocity ratios, volume fractions, and internal heat exchange parameters.     

Forced convection heat transfer inside an anisotropic porous channel with oblique principal axes: Effect of viscous dissipation

An analytical study on laminar and fully developed forced convection heat transfer in a parallel-plate horizontal channel filled with an anisotropic permeability porous medium is performed. The principal axis of the anisotropic porous medium is oriented from 0 to 90 degrees. A constant heat flux is applied on the outer wall of the channel. Both clear (Newtonian) fluid and Darcy viscous dissipations are considered in the energy equation. Directional permeability ratio parameter A¹ is defined to combine both the effect of the dimensionless permeability ratio parameter K¹=(K₁/K₂) and orientation angle φ into one parameter. The effects of the parameter A¹, the Darcy number Da and the modified Brinkman number Br¹ on the heat transfer and fluid flow characteristics in the channels are investigated and presented in graphs. The obtained results show that the parameters A¹, Da and Br¹ have strong effects on the dimensionless normalized velocity and temperature profiles as well as on the Nusselt number. It is found that for a particular value of A¹, called as critical value A_cr¹, the external heat applied to the surface of the channel is balanced by the internal heat generation due to viscous dissipation and the bulk mean temperature approaches the wall temperature. Hence, the Nusselt number approaches infinity for the critical values A_cr¹.     

Forced convection in a bi-disperse porous medium channel: a conjugate problem

Forced convection in a plane channel filled with a saturated bi-disperse porous medium, coupled with conduction in plane slabs bounding the channel, is investigated analytically on the basis of a two-velocity, two-temperature model. It is found that the effect of the finite thermal resistance due to the slabs is to reduce both the heat transfer to the porous medium and the degree of local thermal non-equilibrium. An increase in value of the Péclet number leads to a decrease in the rate of exponential decay in the downstream direction but does not affect the value of a suitably defined Nusselt number. The dependence of Nusselt number on Biot number associated with the boundary slabs, the interphase heat exchange parameter, the interphase thermal conductivity ratio, the interphase effective permeability ratio, and the macroscopic void fraction, is investigated.     

Forced convection in a channel partly occupied by a bidisperse porous medium: Asymmetric case

This paper presents an analytic investigation of forced convection in parallel-plate channel partly occupied by a bidisperse porous medium and partly by a fluid clear of solid material, the distribution being asymmetrical. The walls of the channel are subject to an uniform heat flux; the flow is assumed to be hydrodynamically and thermally fully developed. The layer of a bidisperse porous medium is attached to one of the channel walls; it is modeled utilizing a two-velocity two-temperature formulation using Darcy’s law. The Beavers–Joseph boundary condition is employed at the bidisperse porous medium/clear fluid interface. The dependences of the Nusselt number on a conductivity ratio, a velocity ratio, a volume fraction, internal heat exchange parameter, and the position of the porous-fluid interface are investigated. Both cases of symmetric and asymmetric heating are investigated, which is specified by the asymmetry heating parameter introduced here. For the case of asymmetric heating, a singular behavior of the Nusselt number is found and explained.     

Forced convection in thermally developing region of a channel partially filled with a porous material and optimal porous fraction

Local Nusselt numbers have been obtained on the porous side and the fluid side of the parallel plate channel. Plots to obtain wall heat transfer directly have been presented. Change in wall heat transfer has been examined to establish that the maximum enhancement in heat transfer occurs at a porous fraction of 0.8 at a Darcy number of 0.001. Correspondingly, the maximum enhancement per unit pressure drop occurs at a porous fraction of 0.7. As Darcy number increases, the porous fraction at which the maximum enhancement in heat transfer occurs decreases.     

Linear stability of convection in a vertical channel filled with nanofluid saturated porous medium

The convection in a vertical channel filled with a porous medium saturated by a nanofluid is studied numerically. The effects of Brownian motion and thermophoresis are incorporated in the model used for nanofluid. Also, the flow within the porous region is governed by Brinkman's equation. The generalized eigenvalue problem for the perturbed state is obtained from a normal mode analysis and solved using the Chebyshev spectral collocation method. The Rayleigh number is expressed as an implicit function of the wavenumber with other parameters. The critical wavenumber and the critical Rayleigh number are calculated for different parameters. The preferred modes under critical conditions are detected.     

Free convection in a wavy cavity filled with a porous medium

The steady-state free convection inside a cavity made of two horizontal straight walls and two vertical bent-wavy walls and filled with a fluid-saturated porous medium is numerically investigated in the present paper. The wavy walls are assumed to follow a profile of cosine curve. The horizontal walls are kept adiabatic, while the bent-wavy walls are isothermal but kept at different temperatures. The Darcy and energy equations (in non-dimensional stream function and temperature formulation) are solved numerically using the Galerkin Finite Element Method (FEM). Flow and heat transfer characteristics (isothermal, streamlines and local and average Nusselt numbers) are investigated for some values of the Rayleigh number, cavity aspect ratio and surface waviness parameter. The present results are compared with those reported in the open literature for a square cavity with straight walls. It was found that these results are in excellent agreement.     

Laminar convection in a vertical channel with viscous dissipation and buoyancy effects

The fully developed and laminar convection in a parallel-plate vertical channel is investigated by taking into account both viscous dissipation and buoyancy. Uniform and symmetric temperatures are prescribed at the channel walls. The velocity field is considered as parallel. A perturbation method is employed to solve the momentum balance equation and the energy balance equation. A comparison with the velocity and temperature profiles in the case of laminar forced convection with viscous dissipation is performed in order to point out the effect of buoyancy. The case of convective boundary conditions is also discussed.     

Simulation of turbulent natural convection in a porous cylindrical annulus using a macroscopic two-equation model

This work presents numerical computations for laminar and turbulent natural convection within a horizontal cylindrical annulus filled with a fluid saturated porous medium. Computations covered the range 25 < Ra_m < 500 and 3.2 × 10⁻⁴ > Da > 3.2 × 10⁻⁶ and made use of the finite volume method. The inner and outer walls are maintained at constant but different temperatures. The macroscopic k–ε turbulence model with wall function is used to handle turbulent flows in porous media. First, the turbulence model is switched off and the laminar branch of the solution is found when increasing the Rayleigh number, Ra_m. Subsequently, the turbulence model is included and calculations start at high Ra_m, merging to the laminar branch for a reducing Ra_m. This convergence of results as Ra_m decreases can be seen as an estimate of the so-called laminarization phenomenon. Here, a critical Rayleigh number was not identified and results indicated that when the porosity, Prandtl number, conductivity ratio between the fluid and the solid matrix and Ra_m are kept fixed, the lower the Darcy number, the higher is the difference of the average Nusselt number given by the laminar and turbulent models.     

Free convection in a square cavity filled with a bidisperse porous medium

The classical problem of steady Darcy free convection in a square cavity filled with a porous medium has been extended to the case of a bidisperse porous medium (BDPM) by following the recent model proposed by Nield and Kuznetsov [D.A. Nield, A.V. Kuznetsov, Natural convection about a vertical plate embedded in a bidisperse porous medium, Int. J. Heat Mass Transfer 51 (2008) 1658–1664] and Rees et al. [D.A.S. Rees, D.A. Nield, A.V. Kuznetsov, Vertical free convective boundary-layer flow in a bidisperse porous medium, ASME J. Heat Transfer 130 (2008) 1–9]. The transformed partial differential equations in terms of the dimensionless stream function and temperature are solved numerically using a finite-difference method for some values of the governing parameters when the Rayleigh number Ra is equal to 10² and 10³. Results are presented for the flow field with streamlines, temperature field by isotherms and heat transfer by local and mean Nusselt numbers are presented for both the f- and p-phases. It is found that the most important parameters that influence the fluid flow and heat transfer are the inter-phase heat transfer parameter H and the modified thermal conductivity ratio parameter γ.     

Mixed convection in a square vented enclosure filled with a porous medium

Steady mixed convection flow in a vented enclosure with an isothermal vertical wall and filled with a fluid-saturated porous medium is investigated numerically. The forced flow conditions are imposed by providing an inlet at the bottom surface, and a vent at the top, facing the inlet. The nature and the basic characteristics of the mixed aiding as well as mixed opposing flows that arise are investigated using the Darcy law model. The governing parameters are the Rayleigh number, Péclet number, and the width of the inlet as a fraction of the height of the square enclosure. These parameters are varied over wide ranges and their effect on the heat transfer characteristics is studied in detail.     

Thermally developing forced convection in a porous medium: parallel plate channel with walls at uniform temperature, with axial conduction and viscous dissipation effects

A modified Graetz methodology is applied to investigate the thermal development of forced convection in a parallel plate channel filled by a saturated porous medium, with walls held at uniform temperature, and with the effects of axial conduction and viscous dissipation included. The Brinkman model is employed. The analysis leads to expressions for the local Nusselt number, as a function of the dimensionless longitudinal coordinate and other parameters (Darcy number, Péclet number, Brinkman number).     

Effect of viscous dissipation and radiation on natural convection in a porous medium embedded within vertical annulus

The effect of viscous dissipation and thermal radiation on natural convection in a porous medium embedded within a vertical annular cylinder is investigated. The inner surface of the cylinder is maintained at an isothermal temperature $T_w$ and the outer surface is maintained at ambient temperature $T_{\infty }$. The fluid is assumed to obey the Darcy law. Finite element method is used to solve the partial differential equations governing the fluid flow and heat transfer behavior. The study is focused to investigate the combined effect of viscous dissipation and radiation. Results are presented for different values of the viscous dissipation parameter, radiation parameter, radius ratio, aspect ratio and Rayleigh number. It is observed that the viscous dissipation parameter reduces the average Nusselt number at hot surface. However, the average Nusselt number increases at the cold surface due to increased viscous dissipation parameter.     

Fully developed mixed convection flow of a nanofluid through an inclined channel filled with a porous medium

In this paper, the steady fully developed mixed convection flow of a nanofluid in a channel filled with a porous medium is presented. The walls of the channel are heated by a uniform heat flux and a constant flow rate is considered through the channel. The equations of the problem are made non-dimensional and are observed to depend on the dimensionless parameters, namely the mixed convection parameter λ, the Péclet number Pe, the inclination angle of the channel to the horizontal γ and the nanoparticle volume fraction ?. The effects of these parameters on the fluid and heat transfer characteristics are in detail discussed for three different nanofluids as Cu–water, Al₂O₃–water and TiO₂–water.